Suggested Projects

The following topics have been suggested by REU research mentors, and offer a range of both theoretical and applied problems. The topics are of current interest to mathematicians, but can be understood in a short period of time by a well-trained undergraduate. Mentors may also be willing to work on other topics, depending on the student's and their own interests.

Dynamical wave phenomena requires the understanding of the large time behaviour of partial differential equations. Quantum Mechanics is one such example. Optical and laser systems are of similar nature.Such analysis poses a special challenge both to theorists and computational analysis. The study of various aspects of the nonlinear Schrödinger equation by analytic and numerical methods is proposed.The topic and its nature will be determined according to the interests and experience of the student. Students should know linear algebra and differential equations. Also very useful are: programming, graphics, quantum mechanics, and perhaps complex analysis, advanced calculus/real analysis.

A central topic in mathematics, and more recently in physics, is symmetry, and in particular, how symmetries behave under "combination". To give a simple example, consider a function f(x) in one variable. f(x) is called even if f(x) = f(-x), odd if f(x) = - f(-x). The product of two even (resp. odd) functions is even, while the product of an even and odd function is odd.The goal of this project is to investigate a new combinatorial rule for determining the generalization of this rule to higher dimensions. Let S_n be the symmetric group of permutations of the variables x_1,....,x_n. How do product of functions of x_1,...x_n behave, with respect to the symmetric group, under multiplication? (The case of two variables is equivalent to the case discussed in the previous paragraph.)In 1998 Knutson and Tao gave a new combinatorial rule for the break-up of two functions, which in many respects is more useful than the rule of Littlewood and Richardson. Possible extensions include generalization to more than two functions, and two other types of combinatorial problems such as Schubert calculus for the full flag variety. However, further developments in this field are likely before the summer begins, and the last part of the project will have to be modified somewhat.

The topics are about generating good images in two and three dimensions. In particular, generating cross sectional and 3-dimensional images of objects from data arising from x-rays taken along various lines through the objects. Some real examples include imaging teeth in a human jaw.Students would start out learning the nature of the problem in a very much simplified model situation which can be handled by pencil and paper. Later, the problem will be made more realistic by moving to different presentations of the data acquired in the lab and by taking x-rays at different angles.Students would learn to use various programs like MATHCAD to solve the inverse problems and to begin to analyze the errors that might arise. They would then get real (or well-simulated) data to analyze for their project.

DNA Sequences and Three-dimensional Structure (Wilma Olson)

The DNA base sequence, once thought to be interesting only as a carrier of the genetic blueprint, is now recognized as playing a structural role in modulating the biological activity of genes. Through subtle variations of bending and twisting at the level of neighboring base pairs, the chemical sequence not only generates a higher-order folding of the double helix itself but also produces structural motifs that facilitate the specific binding of proteins and other ligands. The consequent looping and twisting of the DNA assumes an important role in various biological processes.To understand how the genetic code is translated into three-dimensional structures used in biological processes, our group is analyzing the known geometry of DNA bases in chemical structures and developing mathematical models to incorporate the sequence-dependent bending, twisting, and translation of known fragments in DNA molecules of varying lengths and chemical composition.Note that this year's project will build on the successes of previous students who worked with Prof. Olson on this project. See, for example, Lynette Hock's project.